Multiplication Theorem on Probability

Deriving the Multiplication Rule

The definition of conditional probability, $P(A\mid B)=P(A\cap B)/P(B)$ (for $P(B)>0$), can be rearranged by simply multiplying both sides by $P(B)$:

By the same argument starting from $P(B\mid A)=P(A\cap B)/P(A)$ (for $P(A)>0$),

Together these give the multiplication theorem on probability:

provided $P(A)>0$ and $P(B)>0$. In words: the probability that both $A$ and $B$ occur equals the probability that one of them occurs, multiplied by the conditional probability of the other given that the first has occurred.

Extension to Three Events

The rule extends naturally to three events $A$, $B$, $C$ (with $P(A\cap B)>0$):

Each successive factor conditions on everything already assumed to have happened -- this pattern extends to any number of events and is often called the chain rule of probability.

Application: Drawing Without Replacement

The multiplication theorem is the natural tool whenever items are drawn in succession without replacement, since the composition of the pool genuinely changes after each draw, so the probability of the second event depends on the outcome of the first.

Worked illustration. A bag contains $5$ red and $3$ black balls ($8$ balls in all). Two balls are drawn one after another without replacement. Let $R_1$ be the event "the first ball is red" and $R_2$ be the event "the second ball is red." Then $P(R_1)=\tfrac58$. Once the first (red) ball is removed, $7$ balls remain, of which $4$ are red, so $P(R_2\mid R_1)=\tfrac47$. By the multiplication theorem, …